Rate of Change and Slope

Objectives:

• Use the rate of change to solve problems.

• Find the slope of a line.

The word slope (gradient, incline, pitch) is used to describe the measurement of the steepness of a straight line.

Slope

The slope of a line is also known as the rate of change.

Types of Slopes

Positive Slope m = +

Zero Slope m = 0

Undefined Slope

Negative Slope m = -

Slope is a ratio and can be expressed as:

Slope = Vertical Change or Rise or Horizontal Change Run

2 1

2 1x

y y

x

To find the slope in this lesson you must use….

Slope formula2 1

2 1

y ym

x x

Vertical change = 14

P2(x2, y2)

P1(x1, y1)

Horizontal change = 7

Slope = vertical change

horizontal change

Slope = 14 or 2

7

Is the slope positive or negative?

Positive

Find the slope of each line:

run

rise

2

3 run

rise

3

5

Practice

Find the slope of the line that passes through (-1, 4) and (1, -2).Then graph the line.

2 1

2 1x

ym

y

x

1 ( )

2

1

4m

6

23m

Find the slope of the line that passes through each pair of points.

1. (7, 6), (7, 4) 2. (9, 3), (7, 2)

= undefined slope = ½

12

12

xx

yym

12

12

xx

yym

77

64

m

0

2m

97

32

m

2

1

m

Find the slope of the line that passes through each pair of points.

3. (1, 2) (-1, 2) 4. (9, -4) (7, -1)

slope = 0 slope =23

Graph on the coordinate plane m= -3 Passes through

(-1, 2)

Graph on the coordinate plane m= 1/2 Passes through

(2, 3)

Graph on the coordinate plane m= undefined Passes through

(3, 1)

Graphing Equations

● Example: Graph the equation -5x + y = 2Solve for y first.

-5x + y = 2 Add 5x to both sides y = 5x + 2

● The equation y = 5x + 2 is in slope-intercept form, y = mx+b. The y-intercept is 2 and the slope is 5. Graph the line on the coordinate plane.

x

y

Graph y = 5x + 2

Graphing Equations

Graph 4x - 3y = 12

● Solve for y first

4x - 3y =12 Subtract 4x from both sides

-3y = -4x + 12 Divide by -3

y = x + Simplify

y = x – 4● The equation y = x - 4 is in slope-intercept form,

y=mx+b. The y -intercept is -4 and the slope is . Graph the line on the coordinate plane.

Graphing Equations

12-3

43

43

43

-4-3

Graph y = x - 4

x

y

43

Graphing Equations

Find the value of r so that the line through (r, 6) (10, -3) has a slope of 2

3

12

12

xx

yym

r

10

9

2

3

r

10

63

2

3

-3(10 – r) = 2(-9)

-30 + 3r = -18

3r = 12

r = 4

Slope Formula

Let (r, 6) = (x1, y1) and (10, -3) = (x2, y2)

Subtract

Cross Multiply

Use the Distributive Property and simplify

Add 30 to each side and simplify

Divide each side by 3 and simplify

The line goes through (4, 6)

Find the value of r so the line that passes through each pair of points has the given slope.

1. (1, 4) (-1, r); m = 2

2. (r, -6) (5, -8); m = -8

r = 0

r = 4.75

Questions . . . ?

Comments . . . ?

Concerns . . . ?